Chinese Remainder Theorem in Rings (CRT)

Question

In the Chinese Remainder Theorem for Rings we have that the Ideals $I$ and $J$ be comaximal (i.e. $I+J=R$), then one shows that $IJ=I{\cap}J$ and proves the CRT via the first Isomorphism theorem.

Therefore it is clear that $I+J=R$ implies $IJ=I\cap J$.

My question is that

Is there a necessary and a sufficient condition for $I\cap J=IJ$, also is there an example where $I\cap J=IJ$ does not imply $I+J=R$. (One can surely not find such examples in $\mathbb{Z}$ though.)

Edit: Initially I had commented that such examples are not possible in $\mathbb{Z}$ by unknowingly leaving out the $(0)$ ideal ! But that is false as pointed out in the comments.

Answer 1

Is there a necessary and a sufficient condition for $I\cap J=IJ$

Sure: $I\cap J\subseteq IJ$ is necessary and sufficient.

is there an example where $I\cap J=IJ$ does not imply $I+J=R$.

How about $I=J=F_2\times \{0\}$ in the ring $R=F_2\times F_2$. We have $IJ=I\cap J=I\neq R$.

As Arturo already pointed out in comments, if you allow one of the ideals to be zero then you can find an example there too.